Question

If $\left(\sin \alpha \right)x^2-2x+b\ge 2$, for all real values of $x\le 1$ and $\alpha ϵ(0,\frac{\pi }{2})\cup (\frac{\pi }{2},\pi )$, then possible real values of $b$ is /are

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (A), (B)

The correct options are
A $2$
B $3$

$\left(\sin \alpha \right)x^2-2x+b\ge 2\Rightarrow \left(\sin \alpha \right)x^2-2x+b-2\ge 04-4(b-2)\sin \alpha <0\Rightarrow b>\frac{1}{\sin \alpha }+2$
As $-1\le \sin \alpha \le 1\Rightarrow -1\le \frac{1}{\sin \alpha }\le 1\Rightarrow 1\le 2+\frac{1}{\sin \alpha }<3\Rightarrow 1<b\le 3$